Structural stability of adatom islands on fcc(111) transition-metal surfaces

Structural stability of adatom islands on fcc„111… transition-metal surfaces

S. Papadia* and B. PiveteauCommissariat a` l’Energie Atomique, Service de Recherche sur les Surfaces et l’Irradiation de la Matie`re,

Centre d’Etudes de Saclay, F-91 191 Gif sur Yvette, France

D. SpanjaardLaboratoire de Physique des Solides, Universite´ Paris Sud, F-91 405 Orsay, France

M. C. Desjonque`resCommissariat a` l’Energie Atomique, Service de Recherche sur les Surfaces et l’Irradiation de la Matie`re,

Centre d’Etudes de Saclay, F-91 191 Gif sur Yvette, France~Received 29 January 1996!

The energetics of monomer, dimer, and triangular trimer adatom islands on the~111! surface of fcc transi-tion metals of the same chemical species is calculated in the tight-binding scheme as a function of thed bandfilling. Both the adatom and their nearest neighbor substrate atoms are allowed to relax. We investigate allpossible atomic configurations of the system arising from the existence of~a! two types of adsorption sites:normal ~fcc! and fault~hcp! sites, and~b! both types of borders that may exist for triangles of adatoms:A,having~001!, ~010!, and~100! facets andB, having (111̄), (11̄1), and (1̄11) facets. It is found that there is aninversion of relative stability from fault to normal sites when thed band filling is larger than 8.2d electrons peratom for monomers, 7.85 for dimers, 7.5 for trimers of typeA, and 7.6 for trimers of typeB. There is also aninversion of stability for trimers from typeB to typeA when thed band filling is larger than 7.95. All theseresults are in very good agreement with experiments on Ir for which thed band filling is .7.5227.6:monomers and dimers prefer to stick at fault sites while trimers settle at both sites. Furthermore, triangles oftype B are energetically more favored than triangles of typeA. The case of other transition metals is alsodiscussed.@S0163-1829~96!03135-9#

I. INTRODUCTION

The study and characterization of crystal growth1 has,apart from its basic surface science interest, important impli-cations in technology. The process of fabrication of very thinmetal films is commonly used in electronics, optoelectronics,and the recording media industry. The more the growth isunderstood, the more controlled the fabrication processes canbe made and, consequently, a better quality of the deviceperformances can be obtained.

A possible scenario of layer-by-layer growth is the fol-lowing: an atom adsorbs at the surface; it starts diffusinguntil it definitely sticks to a defect, e.g., a step or a vacancy;the step will ‘‘flow’’ and construct the new layer, or therewill be adatom island formations if many adatoms clustertogether before reaching a step site. A ‘‘flow’’ of steps is thedesired, ideal two-dimensional growth, while a growth viaadatom islands may result in a so called three-dimensionalgrowth which gives rise to rougher surfaces. Moreover, it isdesired that the growth also is epitaxial—the new layershould match the substrate and not introduce defects orstacking faults.

For growth on fcc~111! metal surfaces, there are twothreefold adsorption sites: the normal~fcc! site which con-tinues the stacking order of the fcc~111! and the fault~hcp!site which, if occupied by the adatoms during a growth pro-cess, will introduce the hcp stacking order instead and, thus,a stacking fault. Previous calculations2 have shown that epi-taxial monolayers always are energetically favorable on

fcc~111!. However, experiments3 and calculations4 show thatfor a single Ir adatom on Ir~111!, the hcp, or fault, site ispreferred. One may thus expect that for this element a tran-sition from fault to normal site adsorption will occur whenthe adatom island size increases. Furthermore, adatom is-lands are expected to be limited by close-packed atomic rowscorresponding to two different types of microfacets:~001!for typeA and (111̄) for typeB. These edges correspond, inother words, to the type of steps with dense edges existing onthe fcc~111! surface.5

Wang and Ehrlich6 have observed Ir clusters of varioussizes on Ir~111!. They find that there is indeed a transitionfrom preferred fault site adsorption to normal site adsorptionwhen the adclusters consist of at least four atoms and thattriangles with typeB borders are energetically favored overtriangles with typeA borders. Furthermore, experimentalstudies of the equilibrium shape of Pt clusters on Pt~111!have been performed by Michelyet al.7 They observed ir-regular hexagons, which have bothA and B borders, andfound thatB edges are predominant. In view of the large sizeof the considered clusters, the equilibrium shape is domi-nated by the free energies of the steps limiting them. Thecalculation of the corresponding step energies has been pre-sented elsewhere,5 but it is not sure that the results still applyfor small clusters where corner effects may be important. Itis thus of interest to make a systematic investigation of theenergy difference of monomers, dimers, and adatom tri-angles with different edges at the two possible adsorption

PHYSICAL REVIEW B 15 NOVEMBER 1996-IIVOLUME 54, NUMBER 20

540163-1829/96/54~20!/14720~8!/$10.00 14 720 © 1996 The American Physical Society

sites on fcc~111! surfaces of the same chemical species as afunction of thed band filling.

In the following, we proceed with such an investigationwithin a tight-binding model. The paper is organized as fol-lows. In Sec. II, we present the tight-binding model fordelectrons that is used for calculating the binding energies oftransition metal adatom islands. The model is made explicitfor application on adatom island adsorption on fcc~111! sur-faces in Sec. III. In Sec. IV, our numerical results are pre-sented and compared with experiments. Finally, a summaryand our conclusions are given in Sec. V.

II. TIGHT-BINDING MODEL

In a tight-binding semiempirical calculation, the bindingenergyEB of adatoms is usually expressed as the sum of anattractive and a repulsive contribution as8

EB5DEband1DErep, ~1!

whereEband is due to the broadening of valence electroniclevels into a band andErep contains the ion-ion repulsion.TheD indicates that we calculate the variation of band andrepulsive energies relative to a reference system. For ada-toms at a surface, the reference system is a surface and thefree adatoms infinitely apart from each other and from thesurface.

The repulsive energy is described by a Born-Mayer pair-wise potential. Since we limit ourselves to the case where theadatoms and the substrate are transition metals of the samechemical species, it can be written as

Erep5 (i , j, i

Ae2p@~Ri j /R0!21#, ~2!

where A and p are parameters characteristic of a givenmetal,Ri j is the distance between atomsi and j , andR0 isthe equilibrium interatomic distance in the bulk.

Since we are interested in computing band energies fortransition metals, we use a tight-binding Hamiltonian,HTB ,where only thed electrons are explicitly taken into account.To its justification, it has been shown9 that, at least for tran-sition metals not too close to the extremes of the transition-metal series, the cohesive properties at equilibrium are by fardominated by the broadening and the occupancy of the va-lenced levels.

When we deal with impurities, surfaces, etc., the effectiveatomic levels and local densities of states~LDOS’s! are sitedependent. Both quantities are interrelated since perturba-tions in LDOS’s cause a change of the potential and, thus, ofthe atomic level at each site. Consequently, they should becalculated self-consistently. However, in metals, screening ofcharge takes place within an interatomic distance and thechange of atomic levels may be obtained by requiring localcharge neutrality.8

The band contribution,DEband, to the binding energy ofadsorbates can be split into two contributions: one which iscoming from the broadening of the atomic levels of the ada-toms (DEband

ads ), and one which arises from the perturbation ofthe substrate due to the presence of the adatoms (DEband

sub ). If

we call«a* («a) the atomic level of adatoma in the adsorbed~free! state,DEband

ads can be expressed as a sum of contribu-tions of each adatom as

DEbandads 5(

aF10EEF

Ena~E!dE2Nd~«a*2«a!2Nd«aG ,~3!

where na(E) is the LDOS’s of adatoma ~normalized tounity! andNd is the number of valenced electrons of theadatom and substrate atoms since they are assumed to be ofthe same chemical species. The second term in Eq.~3! arisesfrom the double counting of electron-electron interactionsresponsible for the shift («a*2«a), and Eq.~3! simplifies into

DEbandads 5(

aF10EEF

Ena~E!dE2Nd«a* G . ~4!

For an atom at a substrate sitei , the LDOS’sni8(E) isperturbed by the presence of the adsorbate~s! asni8(E)5ni(E)1Dni(E), where ni(E) is the LDOS’s ofatom i in the clean substrate andDni(E) is the perturbationdue to the adatoms.DEband

sub is written as a sum of contribu-tions from each substrate atom:

DEbandsub 5 (

iPsubstrateH10EEF

EDni~E!dE2NdDVi J . ~5!

Here again the second term avoids the double counting of thechange in electron-electron interactions responsible for theshift DVi of the atomic level of substrate atomi due to thepresence of the adatoms.

The matrix elements of the HamiltonianHTB are the hop-ping integrals which have a finite range and are limited hereto nearest neighbors, and the effective atomic levels. Thehopping integrals are completely determined by three hop-ping parametersdds, ddp, ddd ~Ref. 10! and the direc-tion cosines of the vectorRi j connecting sitesi and j . Thevariation of these parameters with distance is taken to beexponential (l5s,p,d):

ddl5ddl0e2q@~Ri j /R021#. ~6!

The local density of states of a given atom is calculatedusing the Green operator:

G~z!51

z2HTB, ~7!

as

ni~E!51

5(l21

plim

«→01

Im^ iluG~E1 i«!u il&,

ni~E!51

5(l21

plim

«→01

ImGiill~E1 i«!,

ni~E!51

5(n,l u^F iluCn&u2d~E2En!, ~8!

whereF il is the atomic orbital of symmetryl, centered atsite i , andCn is the eigenfunction of energyEn .

54 14 721STRUCTURAL STABILITY OF ADATOM ISLANDS ON . . .

The quantity Giill can be expanded as a continued

fraction.8 When the corresponding coefficients are exact upto the leveln, the LDOS’s has 2n exact moments. The moreaccurate a calculation is required to be, the more exact mo-mentsmp

i ,

mpi 5E

2`

1`

Epni~E!dE, ~9!

need to be included.In this work, the LDOS’s of an atom in a system is evalu-

ated by calculating exactlync first levels in a recursionscheme and replacing the remaining part of the continuedfraction by the usual square root terminator which corre-sponds to using the asymptotic values for the remainingcoefficients.8 The parametersddl0 are fitted to band struc-ture calculations at high symmetry points of the Brillouinzone. These parameters are given in Table I, in units of thebandwidth. They are derived for Ir, however, we have keptthem constant as a function of thed band filling since we donot expect that the ratiosddp/dds, andddd/dds vary sub-stantially in other fcc transition metals. The values ofp andq ~Table I! have been derived from Ref. 11. The repulsiveinteraction energy of a pair of atoms is drawn from the bulkequilibrium condition, i.e.:

A52q

6pEbandbulk~Nd!, ~10!

whereEbandbulk(Nd) is the bulk band energy at equilibrium~per

atom!.

III. APPLICATIONS TO ADATOM ISLANDSAT fcc„111… SURFACES

We calculate the binding energies of monomers, dimers,and trimers on the~111! surface of fcc transition metals. Inboth cases, we determine the effective atomic levels andLDOS’s on a cluster containing the adatoms, the surface at-oms which are directly bound to them, and the nearest neigh-bors of the latter. The contributions to the binding energy ofthe adatoms from more distant neighbors are small and canbe neglected. Monomers and dimers can adsorb with onlytwo configurations: normal~fcc! or fault ~hcp! ~Fig. 1!. Forthe trimers, there are six possible configurations of the ada-toms: two linear and four triangular. The adatoms can beadsorbed either at both types of sites, and the triangles canhave edges of typeA @with ~001! microfacets# or B @with

~111̄) microfacets#. We will not study the linear trimerssince, in our model, correlation effects, which tend to stabi-lize atomic configurations with reduced coordinations,12 areexcluded. These effects would probably be important for cal-culating the energy difference between a linear and a trian-gular trimer since, in the latter case, all adatoms have fivenearest neighbors while, for the linear trimer, the adatoms atboth ends have only four nearest neighbors. Consequently,correlation effects will act to favor the linear trimer and thus

TABLE I. Tight-binding parameters used in the calculations aretabulated. The hopping integralsdds0, ddp0, and ddd0 are ex-pressed in units of the bulk bandwidthW. The parametersp andq are dimensionless.

Parameter Value

p 11.40dds0 /W -0.1300ddp0 /W 0.0586ddd0 /W -0.0073

q 3.80

FIG. 1. Possible atomic configurations for monomers, dimersand triangular trimers on fcc~111! surfaces at normal (N) and fault(F) sites. Trimers of typeA with ~001!, ~010!, and~100! microfac-ets have their center above an atom in the surface layer. Trimers oftypeB with (111̄), (11̄1), and (1̄11) microfacets have their centerabove an adsorption site. Four different configurations(AN ,BN ,AF ,BF) are thus possible for triangular trimers.

14 722 54PAPADIA, PIVETEAU, SPANJAARD, AND DESJONQUE`RES

compete with the increase of band energy resulting from theloss of one bond. We consider thus, in the following, onlythe triangular configurations which are shown in Fig. 1.

Two types of atomic relaxation have been considered:first the adatoms are allowed to relax perpendicularly to thesurface, each bond inside the adatom island remaining par-allel to the surface. Second, atomic displacements of surfaceatoms parallel to the surface are allowed in the followingway: each adatom is assumed to give rise to an isotropicdilatation ~or contraction! of the equilateral triangle formedby its three first neighbors in the surface and the displace-ments produced by two different adatoms on the same sur-face atom are supposed to be additive. Under these condi-tions the symmetry of the unrelaxed system is preserved:threefold for monomers and trimers, twofold for dimers.Even though these conditions may seem rather restrictive, wedo think that the main effects of atomic relaxation is ac-counted for in these schemes. Indeed, lateral relaxations ofadatoms are not detected in experiments6 andab initio cal-culations on Al~001! ~Ref. 13! have shown that the displace-ments of the neighbors of the adatoms into the bulk are neg-ligible. The assumption of additive displacements isobviously not rigorous but should be valid as a first approxi-mation. The energy minimization has been carried out withrespect to both displacements. In the following, the perpen-dicular relaxation will be given in the percentage of the bulk~111! interplanar distanced (102Dd/d), while the lateral re-laxation of the neighbors will be referred to the heighth ofthe equilateral triangle formed by three nearest neighbors inthe bulk (102Dh/h).

In order to reach sufficient accuracy in the calculated en-ergies and energy differences of various adatom islands, weusenc510 exact levels in the continued fractions — whichamounts to 20 exact moments. In the following calculationwe have varied thed band filling from Nd56.8 toNd59.2 e2/atom. This range of band fillings includes all fcctransition metals.

It is well known that in bulk crystals there are inversionsof relative stability between the hcp and the fcc stucture as afunction of thed band filling. It has been shown that theseinversions follow from the equality of the first three mo-ments (m1 ,m2 ,m3) of the total density of states in bothstructures.14 Then it can be easily proved that the differencein energy between the two phases cancels at least twice inthe intervalNde@0,10#. In particular, in the range ofd bandfillings considered here, we find, using the tight-binding pa-rameters given in Table I, that there is an inversion of sta-bility from the hcp phase to the fcc one whenNd is largerthan.7.5 e2/atom, in agreement with what is indeed ob-served. In the case of the same cluster of adatoms adsorbedeither at fcc or hcp sites on a~111!fcc surface at the samedistance from the surface, and when the renormalization ofatomic levels is neglected, a similar behavior is also ex-pected since the total densities of states have again the samem1 , m2, m3 in both geometries. Indeed, it is well known15

thatm2 andm3 are, respectively determined by the numberof bonds and of triangular paths on the lattice involving threejumps between nearest neighbors. It has been shown that fora whole layer the inversion of stability occurs at roughly thesame band filling as in the bulk, so that a stacking fault isnever stable at the surface.2 However, the particular value of

band filling at which the inversion occurs may be dependenton the considered clusters~size and geometry!. The samebehavior is also expected when comparingA andB triangles.We will see in the following that the above qualitative trendsremain valid when the model is improved by taking intoaccount the renormalization of the atomic levels of the ada-toms, its neighbors, and of the first layers of the substrate andwhen atomic relaxation is allowed.

IV. RESULTS AND DISCUSSION

A. Single adatoms

In a previous work using exactly the same model,4 wehave been able to explain the inversion of stability betweenfault and normal sites on Ir~111! when thed valence shell ofthe adatom fills. In particular, it was found that an Ir adatomwas more stable at a fault site with an increased stabilitywhen the neighboring surface atoms were allowed to relax.However, no full minimization with respect to both this re-laxation and the vertical relaxation of the adsorbate was car-ried out.

In the present work we limit ourselves to the case of ho-moepitaxy, but a systematic study is performed as a functionof the d band filling. Furthermore the total energy is fullyminimized as a function of the two allowed displacements.The difference of adsorption energy between the fault andthe normal site is shown in Fig. 2. It is seen that an inversionof stability from the hcp to the fcc site occurs atNd58.2 e2/atom. This value is significantly larger than thevalue found when the substrate relaxation is neglected. Thusthe substrate relaxation acts in favor of the hcp site whenNd is smaller than about 8.2e

2/atom. This is clearly relatedto the different behavior of the atomic relaxations found as afunction of thed band filling for the two types of sites~Fig.3!. As expected the normal displacementDd/d of the adatomis directed towards the surface, while the neighbors are lat-erally displaced away from the adatom.

Let us emphasize that the second-moment potential:16

FIG. 2. Difference,DE5EB(F)2EB(N) ~in units of the sub-strate bandwidthW), of the binding energy of a monomer betweenthe fault and the normal adsorption site on fcc~111! transition-metalsurfaces as a function of thed band fillingNd .


Etot5(i

S (j, iAe2p@~Ri j /R0!21#

2BA(jÞ i

e22q@~Ri j /R0!21#D , ~11!

when limited to first nearest neighbors, would lead to atomicdisplacements independent of thed band filling. Indeed, inthis approximation,Eband

bulk(Nd)5BA12 and consequentlyB isproportional toA @see Eq.~10!#. Nevertheless, for the pur-pose of comparison, we have also performed calculationsusing this scheme. Contrary to the above complete calcula-tion, the results are very similar for both sites. This can bequalitatively explained as follows. The second-moment ap-proximation only takes into account the variation of nearestneighbor distances. Consequently, it gives usually the rightsign and order of magnitude of the atomic displacementswhen all nearest distances are increased~or decreased!, as inthe case of normal relaxation of flat surfaces. This is cor-roborated by a calculation with the present model~using 20moments! but allowing vertical displacements of the adatom

only, since this relaxation is then almost independent of thetype of adsorption site and roughly equal to that obtainedwith the second-moment potential. However, in the case ofthe lateral relaxation, such as the substrate relaxation studiedhere, some distances are expanded whereas some others arecompressed and the net effect on the second-moment maynearly cancel. This explains qualitatively why the secondmoment calculation gives almost no lateral relaxation of thesurface atoms. In these conditions angular distortions whichare taken into account in the third and higher moments playthe dominant role. These distortions are obviously dependenton the adsorption site which explains the different behaviorof the lateral relaxation of the adatom neighbors betweenfault and normal sites. In addition, it is clear that if the tri-angle of neighbors expands, the adatom moves towards thesurface to compensate for the increased bond length, so thatfinally both relaxations vary in the same way.

B. Dimers

The difference of adsorption energy per atom between thedimers at fault and normal sites is shown as a function of thed band filling in Fig. 4. The same type of behavior as for themonomer is found. However, on the one hand, this differ-ence is smaller than for the monomer. On the other hand, therange ofd band fillings for which the dimer at fault sites ismore stable than at normal sites is narrower than in the caseof the monomer since the inversion of stability occurs now atNd.7.85 e2/atom. The relaxations corresponding to bothgeometries are given in Fig. 5. It is seen that the relaxationsare larger at fault sites than at normal sites untilNd.8.2 e2/atom. Consequently, as for the adsorption of asingle adatom, relaxations act in favor of the fault site in thelower part of the range ofd band fillings considered here.Actually, when the relaxation of the neighbors of the adatomis neglected, the inversion of stability is found forNd.6.8 e2/atom.

C. Adatom triangles

The four possible types of triangular trimers at anfcc~111! surface which are depicted in Fig. 1 are denoted in

FIG. 3. Atomic relaxations for a monomer at a normal site~solidline! and at a fault site~dashed line! as a function of thed bandfilling Nd : ~a! normal displacement of the adatom towards the sur-face in percentage of the bulk~111! interplanar distanced, ~b!lateral displacement of the neighbors away from the adatom in per-centage of the heighth of the equilateral triangle formed by threenearest neighbors in the bulk.

FIG. 4. Difference,DE5EB(F)2EB(N) ~in units of the sub-strate bandwidthW), of the binding energy per atom of a dimerbetween the fault and the normal adsorption site on fcc~111!transition-metal surfaces as a function of thed band fillingNd .


the following; AN for the typeA triangle at normal sites,AF for the typeA triangle at fault sites,BN for the typeBtriangle at normal sites,BF for the typeB triangle at faultsites.

We have calculated the energy~per atom! of adatom tri-angles in the previously mentioned configurations as a func-tion of thed band filling. The adsorption energy difference~per atom! between fault and normal adsorption sites forAandB triangles is shown in Fig. 6. It exhibits the same quali-tative behavior as for monomers and dimers, but its magni-tude is again decreased. As expected, there is a change ofsign of this difference aroundNd57.5 e2/atom for A tri-angles andNd57.6 e2/atom forB triangles, the normal sitesbeing preferred beyond these values of the band filling. Thusthere is always an inversion of stability between hcp and fccsites whenNd crosses a critical valueNd

c . However, whenthe size of adatom islands increases from 1 to 3 atoms,Nd

c

decreases and, for triangles, it is very close to the criticalvalue ofNd for which the transition hcp-fcc occurs in thebulk, i.e.,Nd

c,bulk57.5 e2/atom.The relaxations obtained for the 4 types of triangles are

shown in Figs. 7 and 8. As in monomers and dimers, therelaxations are larger at fault sites than at normal sites whenNd<8 e2/atom. Thus, in this range ofNd , they act in favorof the fault site by displacingNd

c to larger band fillings butjust enough to obtain a very narrow range of band fillings

aboveNdc,bulk , where the fault and normal triangles have

about the same energy. On the contrary, whenNd<8e2/atom, the relaxations aroundA andB triangles are almostthe same provided that the adatoms occupy the same type ofsites. Consequently, one can infer that the relaxations havelittle influence on the relative stability of trianglesA andB.

FIG. 5. Same caption as Fig. 3 but for the dimer.

FIG. 6. Difference,DE5EB(F)2EB(N) ~in units of the sub-strate bandwidthW), of the binding energy per atom of a trimerbetween the fault and the normal adsorption site on fcc~111!transition-metal surfaces as a function of thed band filling Nd .Solid line:A triangles; dashed line:B triangles.

FIG. 7. Same caption as Fig. 3 but for a trimer of typeA.


In Fig. 9, the energy differences~per atom! of adatomtriangles of typeA and B at normal and fault sites onfcc~111! surfaces are shown. There is a change of sign of thisenergy difference at a band filling around 7.95e2/atom. Formetals with ad band filling less than 7.95e2/atom, the typeB adatom triangle is more stable than the typeA adatomtriangle irrespective of the site. The reverse is true ford bandfillings larger than 7.95e2/atom except for the fault tri-angles where a second inversion of stability fromA to Btriangles occurs atNd.8.85 e2/atom. Let us note that, asexpected from the above discussion on relaxations, the criti-cal value corresponding to the first inversion of stability isnot very sensitive to relaxation. Indeed, when relaxation iscompletely neglected, the critical value is found atNd58.2 e2/atom for both types of sites. Thus it increasesslightly the range of stability ofA triangles. Finally, the en-ergy difference betweenA andB triangles is larger at faultthan at normal sites for low band fillings, the reverse beingobtained at high band fillings.

D. Comparison with experiments

Let us now compare our results with the findings of Wangand Ehrlich6 on Ir clusters. These authors, using field ionmicroscopy, have been able to examine Ir clusters containingfrom 1 to 13 atoms held on the close-packed Ir~111! plane.They have found that in all the clusters examined, atoms sit

in nearest neighbors sites. Furthermore, triangular trimers arefound to be slightly more stable that the linear ones althoughthis difference is at the limit of the capability of the experi-mental method. By mapping all the sites occupied by anadatom after repeated diffusions, they have been able toshow that the fault sites are preferred for the monomers anddimers while normal sites are much favored over fault onesfor tetramers and larger clusters. For triangular trimers bothtypes of sites are roughly equally stable. In addition, tri-anglesB are significantly more stable than trianglesA.

From our calculations, it can be deduced that there exist adomain ofd band [email protected],8.2# for which the hcp siteis more stable than the fcc site for one adatom on a fcc~111!surface of the same chemical species. This domain narrowsvery rapidly when the cluster size increases. Consequently,outside this domain and in the range of stability of bulk fcc,i.e., Nd.8.2 e2/atom, we predict that cluster adatoms sitalways at normal sites irrespective of the size of the cluster.On the contrary [email protected],8.2# a transition from fault tonormal site is expected for clusters containing 2 or 3 atoms.TakingNd.7.527.6 e2/atom for Ir,4 we find, in accordancewith experiments, that Ir monomers and dimers occupy faultsites, whereas fault and normal triangular trimers have al-most the same adsorption energy, butB triangles are signifi-cantly more stable thanA triangles. If we note that Rh hasthe fcc structure and that its band filling must be close to thatof Ir since they belong to the same column in the PeriodicTable, we predict the same kind of behavior for this element.It would be interesting to have experiments performed onthis metal.

Let us now discuss the case of Pt. At larger cluster sizes~several hundreds Å wide!, Michely et al.7 have found thatthe cluster exhibits an irregular hexagonal shape withA andB borders, theB borders being predominant. At these sizes,the shape of the cluster is completely determined by the stepenergies of the borders. The energies of stepsA andB havebeen studied in a previous work,5 neglecting relaxation ef-fects. It was shown that in the range ofd band fillings cor-responding to Pt,A borders were slightly energetically fa-

FIG. 8. Same caption as Fig. 3 but for a trimer of typeB.

FIG. 9. Difference,DE5EB(B)2EB(A) ~in units of the sub-strate bandwidthW!, of the binding energy per atom between typeB and typeA trimers on fcc~111! transition-metal surfaces as afunction of thed band fillingNd . Solid line, trimers at normal sites;dashed line, trimers at fault sites.


vored but the corresponding energy difference was so smallthat it might be overcome by relaxation effects and/or theinfluence ofs electrons. In the present calculation, the dif-ference in energy betweenAN andBN triangles is larger byone order of magnitude even when the relaxations are ne-glected so that it is not clear that the equilibrium geometry ofvery small Pt adatoms islands will be the same as in experi-ments of Michelyet al. on large adatoms islands. Conse-quently, it would be also highly interesting to study the veryfirst stages of growth of Pt on Pt~111!. Experiments on Pd,which is also fcc and in the same column as Pt, would bealso welcome.

V. SUMMARY AND CONCLUSIONS

The binding energies of monomers, dimers, and trimers atnormal and fault sites on fcc~111! surfaces of the samechemical species have been calculated within a tight-bindingmodel ford electrons in order to investigate trends for clus-ter adsorption along the transition-metal series of a fcc struc-ture (d band fillings of 6.829.2 e2/atom!.

Vertical relaxation of the adatom~s! as well as lateral re-laxation of its~their! nearest neighbors have been taken into

account. We have shown that, contrary to the predictions ofthe second-moment potential, these relaxations are quite dif-ferent between the hcp and the fcc sites, and given the reasonfor this discrepancy. This has an important consequencesince the width of the domain ofd band fillings for which thehcp adsorption site on the~111!fcc surface is preferred in-creases.

Our results give a coherent interpretation of the field ionmicroscopy observation of Ir clusters on Ir~111! by Wangand Ehrlich.6 We predict the same trends for the early stagesof the growth of Rh on Rh~111!. It would also be very inter-esting to carry out similar experiments on Pt and Pd to see ifthere is really an inversion of relative stability ofA andBtriangular trimers when going from the Rh and Ir column tothat of Pd and Pt.

ACKNOWLEDGMENTS

One of us~S.P.! acknowledges financial support by E.C.Contract No. ERBCHBICT941049 and the Swedish NaturalScience Research Council~NFR!. We are also indebted to N.Auby for computing advice.

*Present address: FFA, The Aeronautical Research Institute ofSweden, Box 110 21, S-161 21 Bromma, Sweden.

1Kinetics of Ordering and Growth at Surfaces, edited by M. G.Lagally ~Plenum, New York, 1990!; Phys. Today46 ~11!, 24~1993!.

2B. Piveteau, M. C. Desjonque`res, and D. Spanjaard, J. Phys.~France! I 2, 1677~1992!.

3S. C. Wang and G. Ehrlich, J. Chem. Phys.94, 4071 ~1991!;Phys. Rev. Lett.68, 1160~1992!.

4B. Piveteau, M. C. Desjonque`res, and D. Spanjaard, Phys. Rev. B46, 7121~1992!.

5S. Papadia, D. Spanjaard, and M. C. Desjonque`res, Phys. Rev. B53, 4083~1996!.

6S. C. Wang and G. Ehrlich, Surf. Sci.239, 301 ~1990!.7T. Michely, M. Hohage, M. Bott, and G. Comsa, Phys. Rev. Lett.70, 3943~1993!.

8M. C. Desjonque`res and D. Spanjaard,Concepts in Surface Phys-ics ~Springer-Verlag, Berlin, 1993!.

9J. Friedel,The Physics of Metals, edited by J. M. Ziman~Cam-bridge University Press, Cambridge, 1969!, p. 340; C. D. Gelatt,Jr., H. Ehrenreich, and R. E. Watson, Phys. Rev. B15, 1613~1977!; D. G. Pettifor, CALPHAD,1, 305 ~1977!.

10J. C. Slater and G. F. Koster, Phys. Rev. B94, 1498~1954!.11D. Spanjaard and M. C. Desjonque`res, Phys. Rev. B30, 4822

~1984!.12J. P. Bourdin, M. C. Desjonque`res, D. Spanjaard, and J. Friedel,

Surf. Sci.157, L345 ~1985!.13P. J. Feibelman, Phys. Rev. Lett.63, 2488~1989!.14F. Ducastelle,Order and Stability in Alloys~North-Holland, Am-

sterdam, 1991!, Vol. 3, Chap. 6.15F. Cyrot-Lackmann, J. Phys. Chem. Solids29, 1235~1968!.16V. M. Rosato, M. Guillope´, and B. Legrand, Philos. Mag. A59,

321 ~1989!.


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Structural stability of adatom islands on fcc(111) transition-metal surfaces